The first derivatives of all complete elliptic integrals , , and with respect to their variables can also be represented through complete elliptic integrals by the following formulas:

The symbolic derivatives of all complete elliptic integrals , , and with respect to their variables can be represented through Gauss classical or regularized hypergeometric functions by the following formulas:

Integration

The indefinite integrals of all complete elliptic integrals , , and with respect to their variables can be expressed through complete elliptic integrals (or through hypergeometric functions of two variables) by the following formulas: